Chapter 8: Markov Chain Monte Carlo
8.1 Good King Markov and His island kingdom
library(rethinking)Loading required package: rstan
Loading required package: StanHeaders
Loading required package: ggplot2
Registered S3 method overwritten by 'dplyr':
method from
print.rowwise_df
rstan (Version 2.19.2, GitRev: 2e1f913d3ca3)
For execution on a local, multicore CPU with excess RAM we recommend calling
options(mc.cores = parallel::detectCores()).
To avoid recompilation of unchanged Stan programs, we recommend calling
rstan_options(auto_write = TRUE)
Loading required package: parallel
rethinking (Version 1.59)num_weeks <- 1e5
positions <- rep(0, num_weeks)
current <- 10
for (i in 1:num_weeks) {
# record current position
positions[i] <- current
# flip coin to generate proposal
proposal <- current + sample(c(-1, 1), size = 1)
# now make sure he loops arou d the archipelago
if(proposal < 1) proposal <- 10
if(proposal > 10) proposal <- 1
# move?
prob_move <- proposal/current
current <- ifelse(runif(1) < prob_move, proposal, current)
}week <- 1:100
plot(week, positions[1:100], col = rangi2)
visits <- as.vector(table(positions))
plot(1:10, visits)
8.3 Easy HMC: map2stan
library(rethinking)
# https://github.com/stan-dev/rstan/wiki/RStan-Getting-Started
data(rugged)
d <- rugged
d$log_gdp <- log(d$rgdppc_2000)
dd <- d[complete.cases(d$rgdppc_2000), ]m8.1 <- map(
alist(
log_gdp ~ dnorm(mu, sigma),
mu <- a + bR*rugged + bA*cont_africa + bAR*rugged*cont_africa,
a ~ dnorm(0, 100),
bR ~ dnorm(0, 10),
bA ~ dnorm(0, 10),
bAR ~ dnorm(0, 10),
sigma ~ dunif(0, 10)
),
data = dd
)
precis(m8.1)dd.trim <- dd[,c("log_gdp", "rugged", "cont_africa")]
str(dd.trim)'data.frame': 170 obs. of 3 variables:
$ log_gdp : num 7.49 8.22 9.93 9.41 7.79 ...
$ rugged : num 0.858 3.427 0.769 0.775 2.688 ...
$ cont_africa: int 1 0 0 0 0 0 0 0 0 1 ...m8.1stan <- map2stan(
alist(
log_gdp ~ dnorm(mu, sigma),
mu <- a + bR*rugged + bA*cont_africa + bAR*rugged*cont_africa,
a ~ dnorm(0, 100),
bR ~ dnorm(0, 10),
bA ~ dnorm(0, 10),
bAR ~ dnorm(0, 10),
sigma ~ dcauchy(0, 2)
),
data = dd.trim
)
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 6.5e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.65 seconds.
Chain 1: Adjust your expectations accordingly!
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Chain 1:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3.7e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.37 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 1.1e-05 seconds (Warm-up)
Chain 1: 0.001957 seconds (Sampling)
Chain 1: 0.001968 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
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[ 1000 / 1000 ]precis(m8.1stan)NAm8.1stan_4chains <- map2stan(m8.1stan, chains = 4, cores = 4)starting worker pid=42718 on localhost:11424 at 13:11:02.055
starting worker pid=42732 on localhost:11424 at 13:11:02.358
starting worker pid=42746 on localhost:11424 at 13:11:02.691
starting worker pid=42760 on localhost:11424 at 13:11:02.996
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 4.3e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.43 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
[1] "Error in sampler$call_sampler(args_list[[i]]) : "
[2] " c++ exception (unknown reason)" error occurred during calling the sampler; sampling not done
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 2).
Chain 2:
Chain 2: Gradient evaluation took 4.6e-05 seconds
Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.46 seconds.
Chain 2: Adjust your expectations accordingly!
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SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 3).
Chain 3:
Chain 3: Gradient evaluation took 4.2e-05 seconds
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SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 4).
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Chain 4: some chains had errors; consider specifying chains = 1 to debughere are whatever error messages were returned[[1]]
Stan model 'log_gdp ~ dnorm(mu, sigma)' does not contain samples.
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3.4e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.34 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
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Chain 1:
Chain 1: Elapsed Time: 2e-06 seconds (Warm-up)
Chain 1: 0.000937 seconds (Sampling)
Chain 1: 0.000939 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
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[ 3000 / 3000 ]precis(m8.1stan_4chains)# To pull out the samples
post <- extract.samples(m8.1stan)
str(post)List of 5
$ a : num [1:1000(1d)] 9.47 9.29 9.2 9 9 ...
$ bR : num [1:1000(1d)] -0.336 -0.298 -0.242 -0.196 -0.145 ...
$ bA : num [1:1000(1d)] -2.4 -2.16 -1.82 -1.84 -1.64 ...
$ bAR : num [1:1000(1d)] 0.724 0.419 0.519 0.298 0.424 ...
$ sigma: num [1:1000(1d)] 1.008 0.924 0.852 0.879 0.908 ...pairs(post)
pairs(m8.1stan)
show(m8.1stan)map2stan model fit
1000 samples from 1 chain
Formula:
log_gdp ~ dnorm(mu, sigma)
mu <- a + bR * rugged + bA * cont_africa + bAR * rugged * cont_africa
a ~ dnorm(0, 100)
bR ~ dnorm(0, 10)
bA ~ dnorm(0, 10)
bAR ~ dnorm(0, 10)
sigma ~ dcauchy(0, 2)
Log-likelihood at expected values: -229.43
Deviance: 458.86
DIC: 468.43
Effective number of parameters (pD): 4.78
WAIC (SE): 468.9 (14.8)
pWAIC: 4.93 plot(m8.1stan, window = c(80, 2000))
# plot(m8.1stan)stancode(m8.1stan)data{
int<lower=1> N;
real log_gdp[N];
real rugged[N];
int cont_africa[N];
}
parameters{
real a;
real bR;
real bA;
real bAR;
real<lower=0> sigma;
}
model{
vector[N] mu;
sigma ~ cauchy( 0 , 2 );
bAR ~ normal( 0 , 10 );
bA ~ normal( 0 , 10 );
bR ~ normal( 0 , 10 );
a ~ normal( 0 , 100 );
for ( i in 1:N ) {
mu[i] = a + bR * rugged[i] + bA * cont_africa[i] + bAR * rugged[i] * cont_africa[i];
}
log_gdp ~ normal( mu , sigma );
}
generated quantities{
vector[N] mu;
real dev;
dev = 0;
for ( i in 1:N ) {
mu[i] = a + bR * rugged[i] + bA * cont_africa[i] + bAR * rugged[i] * cont_africa[i];
}
dev = dev + (-2)*normal_lpdf( log_gdp | mu , sigma );
}8.4 Care and feeding of your Markov chain
8.4.1 How many samples do you need?
8.4.3 Taming a wild chain
y <- c(-1,1)
m8.2 <- map2stan(
alist(
y ~ dnorm(mu, sigma),
mu <- alpha
),
data = list(y=y),
start = list(alpha=0, sigma = 1),
chains = 2, iter = 4000, warmup = 1000
)
SAMPLING FOR MODEL 'y ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 8e-06 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.08 seconds.
Chain 1: Adjust your expectations accordingly!
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Chain 1:
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Chain 1:
SAMPLING FOR MODEL 'y ~ dnorm(mu, sigma)' NOW (CHAIN 2).
Chain 2:
Chain 2: Gradient evaluation took 5e-06 seconds
Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
Chain 2: Adjust your expectations accordingly!
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Chain 2:
Chain 2: Elapsed Time: 0.192994 seconds (Warm-up)
Chain 2: 0.886279 seconds (Sampling)
Chain 2: 1.07927 seconds (Total)
Chain 2: There were 530 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupThere were 189 transitions after warmup that exceeded the maximum treedepth. Increase max_treedepth above 10. See
http://mc-stan.org/misc/warnings.html#maximum-treedepth-exceededThere were 2 chains where the estimated Bayesian Fraction of Missing Information was low. See
http://mc-stan.org/misc/warnings.html#bfmi-lowExamine the pairs() plot to diagnose sampling problems
The largest R-hat is 1.46, indicating chains have not mixed.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#r-hatBulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-ess
SAMPLING FOR MODEL 'y ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 5e-06 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 2e-06 seconds (Warm-up)
Chain 1: 0.000767 seconds (Sampling)
Chain 1: 0.000769 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
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[ 6000 / 6000 ]There were 530 divergent iterations during sampling.
Check the chains (trace plots, n_eff, Rhat) carefully to ensure they are valid.precis(m8.2)There were 530 divergent iterations during sampling.
Check the chains (trace plots, n_eff, Rhat) carefully to ensure they are valid.plot(m8.2)
m8.3 <- map2stan(
alist(
y ~ dnorm(mu, sigma),
mu <- alpha,
alpha ~ dnorm(1, 10),
sigma ~ dcauchy(0, 1)
),
data = list(y = y),
start = list(alpha = 0, sigma = 1),
chains = 2, iter = 4000, warmup = 1000
)
precis(m8.3)m8.3 <- map2stan(
alist(
y ~ dnorm(mu, sigma),
mu <- alpha,
alpha ~ dnorm(1, 10),
sigma ~ dcauchy(0, 1)
),
data = list(y = y),
start = list(alpha = 0, sigma = 1),
chains = 2, iter = 4000, warmup = 1000
)
SAMPLING FOR MODEL 'y ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 1e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.1 seconds.
Chain 1: Adjust your expectations accordingly!
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Chain 1:
SAMPLING FOR MODEL 'y ~ dnorm(mu, sigma)' NOW (CHAIN 2).
Chain 2:
Chain 2: Gradient evaluation took 7e-06 seconds
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Chain 2: Adjust your expectations accordingly!
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Chain 2:
SAMPLING FOR MODEL 'y ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3.1e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.31 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
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Chain 1:
Chain 1: Elapsed Time: 3e-06 seconds (Warm-up)
Chain 1: 0.000728 seconds (Sampling)
Chain 1: 0.000731 seconds (Total)
Chain 1: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
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[ 6000 / 6000 ]precis(m8.3)y <- rcauchy(1e4, 0, 5)
mu <- sapply(1:length(y), function(i) sum(y[1:i])/i)
plot(mu, type = "l")
8.4.4 Non-identifiable parameters
y <- rnorm(100, mean = 0, sd = 1)m8.4 <- map2stan(
alist(
y ~ dnorm(mu, sigma),
mu <- a1 + a2,
sigma ~ dcauchy(0, 1)
),
data = list(y = y), start = list(a1 = 0, a2 = 0, sigma = 1),
chains = 2, iter = 4000, warmup = 1000)
SAMPLING FOR MODEL 'y ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3.8e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.38 seconds.
Chain 1: Adjust your expectations accordingly!
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Chain 1:
Chain 1: Elapsed Time: 3.84316 seconds (Warm-up)
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Chain 1: 17.1454 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'y ~ dnorm(mu, sigma)' NOW (CHAIN 2).
Chain 2:
Chain 2: Gradient evaluation took 9e-06 seconds
Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.09 seconds.
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Chain 2: Iteration: 4000 / 4000 [100%] (Sampling)
Chain 2:
Chain 2: Elapsed Time: 3.52399 seconds (Warm-up)
Chain 2: 11.8959 seconds (Sampling)
Chain 2: 15.4199 seconds (Total)
Chain 2: There were 4949 transitions after warmup that exceeded the maximum treedepth. Increase max_treedepth above 10. See
http://mc-stan.org/misc/warnings.html#maximum-treedepth-exceededExamine the pairs() plot to diagnose sampling problems
The largest R-hat is 2.33, indicating chains have not mixed.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#r-hatBulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-ess
SAMPLING FOR MODEL 'y ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 9e-06 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.09 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 2e-06 seconds (Warm-up)
Chain 1: 0.000736 seconds (Sampling)
Chain 1: 0.000738 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
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[ 6000 / 6000 ]precis(m8.4)plot(m8.3)
m8.5 <- map2stan(
alist(
y ~ dnorm( mu , sigma ) ,
mu <- a1 + a2 ,
a1 ~ dnorm( 0 , 10 ) ,
a2 ~ dnorm( 0 , 10 ) ,
sigma ~ dcauchy( 0 , 1 )
) ,
data=list(y=y) , start=list(a1=0,a2=0,sigma=1) ,
chains=2 , iter=4000 , warmup=1000 )recompiling to avoid crashing R session
SAMPLING FOR MODEL 'y ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 1.5e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.15 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 4000 [ 0%] (Warmup)
Chain 1: Iteration: 400 / 4000 [ 10%] (Warmup)
Chain 1: Iteration: 800 / 4000 [ 20%] (Warmup)
Chain 1: Iteration: 1001 / 4000 [ 25%] (Sampling)
Chain 1: Iteration: 1400 / 4000 [ 35%] (Sampling)
Chain 1: Iteration: 1800 / 4000 [ 45%] (Sampling)
Chain 1: Iteration: 2200 / 4000 [ 55%] (Sampling)
Chain 1: Iteration: 2600 / 4000 [ 65%] (Sampling)
Chain 1: Iteration: 3000 / 4000 [ 75%] (Sampling)
Chain 1: Iteration: 3400 / 4000 [ 85%] (Sampling)
Chain 1: Iteration: 3800 / 4000 [ 95%] (Sampling)
Chain 1: Iteration: 4000 / 4000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.992397 seconds (Warm-up)
Chain 1: 2.59583 seconds (Sampling)
Chain 1: 3.58823 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'y ~ dnorm(mu, sigma)' NOW (CHAIN 2).
Chain 2:
Chain 2: Gradient evaluation took 2.6e-05 seconds
Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.26 seconds.
Chain 2: Adjust your expectations accordingly!
Chain 2:
Chain 2:
Chain 2: Iteration: 1 / 4000 [ 0%] (Warmup)
Chain 2: Iteration: 400 / 4000 [ 10%] (Warmup)
Chain 2: Iteration: 800 / 4000 [ 20%] (Warmup)
Chain 2: Iteration: 1001 / 4000 [ 25%] (Sampling)
Chain 2: Iteration: 1400 / 4000 [ 35%] (Sampling)
Chain 2: Iteration: 1800 / 4000 [ 45%] (Sampling)
Chain 2: Iteration: 2200 / 4000 [ 55%] (Sampling)
Chain 2: Iteration: 2600 / 4000 [ 65%] (Sampling)
Chain 2: Iteration: 3000 / 4000 [ 75%] (Sampling)
Chain 2: Iteration: 3400 / 4000 [ 85%] (Sampling)
Chain 2: Iteration: 3800 / 4000 [ 95%] (Sampling)
Chain 2: Iteration: 4000 / 4000 [100%] (Sampling)
Chain 2:
Chain 2: Elapsed Time: 0.905778 seconds (Warm-up)
Chain 2: 3.0553 seconds (Sampling)
Chain 2: 3.96108 seconds (Total)
Chain 2:
SAMPLING FOR MODEL 'y ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 1e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.1 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 4e-06 seconds (Warm-up)
Chain 1: 0.001241 seconds (Sampling)
Chain 1: 0.001245 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
Constructing posterior predictions[ 600 / 6000 ]
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8.6 Practice
Medium
8M1
plot(m8.5)
data(rugged)
d <- rugged
d$log_gdp <- log(d$rgdppc_2000)
dd <- d[ complete.cases(d$rgdppc_2000) , ]
dd.trim <- dd[ , c("log_gdp","rugged","cont_africa") ]
m8m1.ch <- map2stan(
alist(
log_gdp ~ dnorm( mu , sigma ) ,
mu <- a + bR*rugged + bA*cont_africa + bAR*rugged*cont_africa ,
a ~ dnorm(0,100),
bR ~ dnorm(0,10),
bA ~ dnorm(0,10),
bAR ~ dnorm(0,10),
sigma ~ dcauchy(0,2)
) ,
data=dd.trim )recompiling to avoid crashing R session
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3.6e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.36 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
Chain 1: Iteration: 200 / 2000 [ 10%] (Warmup)
Chain 1: Iteration: 400 / 2000 [ 20%] (Warmup)
Chain 1: Iteration: 600 / 2000 [ 30%] (Warmup)
Chain 1: Iteration: 800 / 2000 [ 40%] (Warmup)
Chain 1: Iteration: 1000 / 2000 [ 50%] (Warmup)
Chain 1: Iteration: 1001 / 2000 [ 50%] (Sampling)
Chain 1: Iteration: 1200 / 2000 [ 60%] (Sampling)
Chain 1: Iteration: 1400 / 2000 [ 70%] (Sampling)
Chain 1: Iteration: 1600 / 2000 [ 80%] (Sampling)
Chain 1: Iteration: 1800 / 2000 [ 90%] (Sampling)
Chain 1: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.323197 seconds (Warm-up)
Chain 1: 0.332955 seconds (Sampling)
Chain 1: 0.656152 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 0.00012 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 1.2 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 3e-06 seconds (Warm-up)
Chain 1: 0.00085 seconds (Sampling)
Chain 1: 0.000853 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
Constructing posterior predictions[ 100 / 1000 ]
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m8.m1.un <- map2stan(
alist(
log_gdp ~ dnorm( mu , sigma ) ,
mu <- a + bR*rugged + bA*cont_africa + bAR*rugged*cont_africa ,
a ~ dnorm(0,100),
bR ~ dnorm(0,10),
bA ~ dnorm(0,10),
bAR ~ dnorm(0,10),
sigma ~ dunif(0,10)
) ,
data=dd.trim )
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 6e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.6 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
Chain 1: Iteration: 200 / 2000 [ 10%] (Warmup)
Chain 1: Iteration: 400 / 2000 [ 20%] (Warmup)
Chain 1: Iteration: 600 / 2000 [ 30%] (Warmup)
Chain 1: Iteration: 800 / 2000 [ 40%] (Warmup)
Chain 1: Iteration: 1000 / 2000 [ 50%] (Warmup)
Chain 1: Iteration: 1001 / 2000 [ 50%] (Sampling)
Chain 1: Iteration: 1200 / 2000 [ 60%] (Sampling)
Chain 1: Iteration: 1400 / 2000 [ 70%] (Sampling)
Chain 1: Iteration: 1600 / 2000 [ 80%] (Sampling)
Chain 1: Iteration: 1800 / 2000 [ 90%] (Sampling)
Chain 1: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.276899 seconds (Warm-up)
Chain 1: 0.302603 seconds (Sampling)
Chain 1: 0.579502 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 5.2e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.52 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 3e-06 seconds (Warm-up)
Chain 1: 0.000935 seconds (Sampling)
Chain 1: 0.000938 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
Constructing posterior predictions[ 100 / 1000 ]
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[ 1000 / 1000 ]precis(m8.m1.un)pairs(m8.m1.un)
m8.m1.exp <- map2stan(
alist(
log_gdp ~ dnorm( mu , sigma ) ,
mu <- a + bR*rugged + bA*cont_africa + bAR*rugged*cont_africa ,
a ~ dnorm(0,100),
bR ~ dnorm(0,10),
bA ~ dnorm(0,10),
bAR ~ dnorm(0,10),
sigma ~ dexp(1)
) ,
data=dd.trim )
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3.4e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.34 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
Chain 1: Iteration: 200 / 2000 [ 10%] (Warmup)
Chain 1: Iteration: 400 / 2000 [ 20%] (Warmup)
Chain 1: Iteration: 600 / 2000 [ 30%] (Warmup)
Chain 1: Iteration: 800 / 2000 [ 40%] (Warmup)
Chain 1: Iteration: 1000 / 2000 [ 50%] (Warmup)
Chain 1: Iteration: 1001 / 2000 [ 50%] (Sampling)
Chain 1: Iteration: 1200 / 2000 [ 60%] (Sampling)
Chain 1: Iteration: 1400 / 2000 [ 70%] (Sampling)
Chain 1: Iteration: 1600 / 2000 [ 80%] (Sampling)
Chain 1: Iteration: 1800 / 2000 [ 90%] (Sampling)
Chain 1: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.293922 seconds (Warm-up)
Chain 1: 0.350508 seconds (Sampling)
Chain 1: 0.64443 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 0.000136 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 1.36 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 3e-06 seconds (Warm-up)
Chain 1: 0.000764 seconds (Sampling)
Chain 1: 0.000767 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
Constructing posterior predictions[ 100 / 1000 ]
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I can’t tell any difference.
8M2
# plot sigma densities
sigma.ch <- extract.samples( m8m1.ch)$sigma
sigma.un <- extract.samples( m8.m1.un)$sigma
sigma.ex <- extract.samples( m8.m1.exp)$sigma
par(mfrow=c(1,1))
dens(sigma.ch, xlim=c(0.7, 1.2), col='red')
dens(sigma.un, add=T, col='blue')dens(sigma.ex, add=T)
# fit models with a cauchy prior on sigma for varying scale parameter values
m8.2.cauchy.10 <- map2stan(
alist(
log_gdp ~ dnorm( mu , sigma ) ,
mu <- a + bR*rugged + bA*cont_africa + bAR*rugged*cont_africa ,
a ~ dnorm(0, 100),
bR ~ dnorm(0, 10),
bA ~ dnorm(0, 10),
bAR ~ dnorm(0, 10),
sigma ~ dcauchy(0, 10)
), data=dd.trim )
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 4.1e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.41 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
Chain 1: Iteration: 200 / 2000 [ 10%] (Warmup)
Chain 1: Iteration: 400 / 2000 [ 20%] (Warmup)
Chain 1: Iteration: 600 / 2000 [ 30%] (Warmup)
Chain 1: Iteration: 800 / 2000 [ 40%] (Warmup)
Chain 1: Iteration: 1000 / 2000 [ 50%] (Warmup)
Chain 1: Iteration: 1001 / 2000 [ 50%] (Sampling)
Chain 1: Iteration: 1200 / 2000 [ 60%] (Sampling)
Chain 1: Iteration: 1400 / 2000 [ 70%] (Sampling)
Chain 1: Iteration: 1600 / 2000 [ 80%] (Sampling)
Chain 1: Iteration: 1800 / 2000 [ 90%] (Sampling)
Chain 1: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.293972 seconds (Warm-up)
Chain 1: 0.318073 seconds (Sampling)
Chain 1: 0.612045 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3.5e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.35 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 1.1e-05 seconds (Warm-up)
Chain 1: 0.008013 seconds (Sampling)
Chain 1: 0.008024 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
Constructing posterior predictions[ 100 / 1000 ]
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[ 1000 / 1000 ]m8.2.cauchy.1 <- map2stan(
alist(
log_gdp ~ dnorm( mu , sigma ) ,
mu <- a + bR*rugged + bA*cont_africa + bAR*rugged*cont_africa ,
a ~ dnorm(0, 100),
bR ~ dnorm(0, 10),
bA ~ dnorm(0, 10),
bAR ~ dnorm(0, 10),
sigma ~ dcauchy(0, 1)
), data=dd.trim )
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3.1e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.31 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
Chain 1: Iteration: 200 / 2000 [ 10%] (Warmup)
Chain 1: Iteration: 400 / 2000 [ 20%] (Warmup)
Chain 1: Iteration: 600 / 2000 [ 30%] (Warmup)
Chain 1: Iteration: 800 / 2000 [ 40%] (Warmup)
Chain 1: Iteration: 1000 / 2000 [ 50%] (Warmup)
Chain 1: Iteration: 1001 / 2000 [ 50%] (Sampling)
Chain 1: Iteration: 1200 / 2000 [ 60%] (Sampling)
Chain 1: Iteration: 1400 / 2000 [ 70%] (Sampling)
Chain 1: Iteration: 1600 / 2000 [ 80%] (Sampling)
Chain 1: Iteration: 1800 / 2000 [ 90%] (Sampling)
Chain 1: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.302947 seconds (Warm-up)
Chain 1: 0.274696 seconds (Sampling)
Chain 1: 0.577643 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 0.000126 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 1.26 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 2e-06 seconds (Warm-up)
Chain 1: 0.000831 seconds (Sampling)
Chain 1: 0.000833 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
Constructing posterior predictions[ 100 / 1000 ]
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[ 1000 / 1000 ]m8.2.cauchy.point.1 <- map2stan(
alist(
log_gdp ~ dnorm( mu , sigma ) ,
mu <- a + bR*rugged + bA*cont_africa + bAR*rugged*cont_africa ,
a ~ dnorm(0, 100),
bR ~ dnorm(0, 10),
bA ~ dnorm(0, 10),
bAR ~ dnorm(0, 10),
sigma ~ dcauchy(0, .1)
), data=dd.trim )recompiling to avoid crashing R session
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3.1e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.31 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
Chain 1: Iteration: 200 / 2000 [ 10%] (Warmup)
Chain 1: Iteration: 400 / 2000 [ 20%] (Warmup)
Chain 1: Iteration: 600 / 2000 [ 30%] (Warmup)
Chain 1: Iteration: 800 / 2000 [ 40%] (Warmup)
Chain 1: Iteration: 1000 / 2000 [ 50%] (Warmup)
Chain 1: Iteration: 1001 / 2000 [ 50%] (Sampling)
Chain 1: Iteration: 1200 / 2000 [ 60%] (Sampling)
Chain 1: Iteration: 1400 / 2000 [ 70%] (Sampling)
Chain 1: Iteration: 1600 / 2000 [ 80%] (Sampling)
Chain 1: Iteration: 1800 / 2000 [ 90%] (Sampling)
Chain 1: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.350553 seconds (Warm-up)
Chain 1: 0.302521 seconds (Sampling)
Chain 1: 0.653074 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 2.3e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.23 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 3e-06 seconds (Warm-up)
Chain 1: 0.000901 seconds (Sampling)
Chain 1: 0.000904 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
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[ 1000 / 1000 ]8M3
# fit models with an exponential prior on sigma for varying scale parameter values
m8.2.exp.10 <- map2stan(
alist(
log_gdp ~ dnorm( mu , sigma ) ,
mu <- a + bR*rugged + bA*cont_africa + bAR*rugged*cont_africa ,
a ~ dnorm(0, 100),
bR ~ dnorm(0, 10),
bA ~ dnorm(0, 10),
bAR ~ dnorm(0, 10),
sigma ~ dexp(10)
), data=dd.trim )
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3.1e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.31 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
Chain 1: Iteration: 200 / 2000 [ 10%] (Warmup)
Chain 1: Iteration: 400 / 2000 [ 20%] (Warmup)
Chain 1: Iteration: 600 / 2000 [ 30%] (Warmup)
Chain 1: Iteration: 800 / 2000 [ 40%] (Warmup)
Chain 1: Iteration: 1000 / 2000 [ 50%] (Warmup)
Chain 1: Iteration: 1001 / 2000 [ 50%] (Sampling)
Chain 1: Iteration: 1200 / 2000 [ 60%] (Sampling)
Chain 1: Iteration: 1400 / 2000 [ 70%] (Sampling)
Chain 1: Iteration: 1600 / 2000 [ 80%] (Sampling)
Chain 1: Iteration: 1800 / 2000 [ 90%] (Sampling)
Chain 1: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.290757 seconds (Warm-up)
Chain 1: 0.258024 seconds (Sampling)
Chain 1: 0.548781 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 8.7e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.87 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 4e-06 seconds (Warm-up)
Chain 1: 0.001088 seconds (Sampling)
Chain 1: 0.001092 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
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[ 1000 / 1000 ]m8.2.exp.1 <- map2stan(
alist(
log_gdp ~ dnorm( mu , sigma ) ,
mu <- a + bR*rugged + bA*cont_africa + bAR*rugged*cont_africa ,
a ~ dnorm(0, 100),
bR ~ dnorm(0, 10),
bA ~ dnorm(0, 10),
bAR ~ dnorm(0, 10),
sigma ~ dexp(1)
), data=dd.trim )recompiling to avoid crashing R session
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 0.000205 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 2.05 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
Chain 1: Iteration: 200 / 2000 [ 10%] (Warmup)
Chain 1: Iteration: 400 / 2000 [ 20%] (Warmup)
Chain 1: Iteration: 600 / 2000 [ 30%] (Warmup)
Chain 1: Iteration: 800 / 2000 [ 40%] (Warmup)
Chain 1: Iteration: 1000 / 2000 [ 50%] (Warmup)
Chain 1: Iteration: 1001 / 2000 [ 50%] (Sampling)
Chain 1: Iteration: 1200 / 2000 [ 60%] (Sampling)
Chain 1: Iteration: 1400 / 2000 [ 70%] (Sampling)
Chain 1: Iteration: 1600 / 2000 [ 80%] (Sampling)
Chain 1: Iteration: 1800 / 2000 [ 90%] (Sampling)
Chain 1: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.309351 seconds (Warm-up)
Chain 1: 0.279717 seconds (Sampling)
Chain 1: 0.589068 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 0.000138 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 1.38 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 4e-06 seconds (Warm-up)
Chain 1: 0.000928 seconds (Sampling)
Chain 1: 0.000932 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
Constructing posterior predictions[ 100 / 1000 ]
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[ 1000 / 1000 ]m8.2.exp.point.1 <- map2stan(
alist(
log_gdp ~ dnorm( mu , sigma ) ,
mu <- a + bR*rugged + bA*cont_africa + bAR*rugged*cont_africa ,
a ~ dnorm(0, 100),
bR ~ dnorm(0, 10),
bA ~ dnorm(0, 10),
bAR ~ dnorm(0, 10),
sigma ~ dexp(.1)
), data=dd.trim )
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.3 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
Chain 1: Iteration: 200 / 2000 [ 10%] (Warmup)
Chain 1: Iteration: 400 / 2000 [ 20%] (Warmup)
Chain 1: Iteration: 600 / 2000 [ 30%] (Warmup)
Chain 1: Iteration: 800 / 2000 [ 40%] (Warmup)
Chain 1: Iteration: 1000 / 2000 [ 50%] (Warmup)
Chain 1: Iteration: 1001 / 2000 [ 50%] (Sampling)
Chain 1: Iteration: 1200 / 2000 [ 60%] (Sampling)
Chain 1: Iteration: 1400 / 2000 [ 70%] (Sampling)
Chain 1: Iteration: 1600 / 2000 [ 80%] (Sampling)
Chain 1: Iteration: 1800 / 2000 [ 90%] (Sampling)
Chain 1: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.278618 seconds (Warm-up)
Chain 1: 0.265619 seconds (Sampling)
Chain 1: 0.544237 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 2.2e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.22 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 2e-06 seconds (Warm-up)
Chain 1: 0.000949 seconds (Sampling)
Chain 1: 0.000951 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
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[ 1000 / 1000 ]# plot the posterior distribution for sigma under the cauchy priors
sigma.cauchy.10 <- extract.samples(m8.2.cauchy.10, pars="sigma")
sigma.cauchy.1 <- extract.samples(m8.2.cauchy.1, pars="sigma")
sigma.cauchy.point.1 <- extract.samples(m8.2.cauchy.point.1, pars="sigma")
dens(sigma.cauchy.10[[1]], xlab="sigma", col="red")
dens(sigma.cauchy.1[[1]], add=TRUE, col="blue")dens(sigma.cauchy.point.1[[1]], add=TRUE, col="green")
# plot the posterior distribution for sigma under the exponential priors
sigma.exp.10 <- extract.samples(m8.2.exp.10, pars="sigma")
sigma.exp.1 <- extract.samples(m8.2.exp.1, pars="sigma")
sigma.exp.point.1 <- extract.samples(m8.2.exp.point.1, pars="sigma")
dens(sigma.exp.10[[1]], xlab="sigma", col="red")
dens(sigma.exp.1[[1]], add=TRUE, col="blue")dens(sigma.exp.point.1[[1]], add=TRUE, col="green")
Alternative approach:
# estimate the terrain ruggedness model with varying values for warmup
m <- map2stan(
alist(
log_gdp ~ dnorm( mu , sigma ) ,
mu <- a + bR*rugged + bA*cont_africa + bAR*rugged*cont_africa ,
a ~ dnorm(0,100),
bR ~ dnorm(0,10),
bA ~ dnorm(0,10),
bAR ~ dnorm(0,10),
sigma ~ dcauchy(0,2)
), data=dd.trim )recompiling to avoid crashing R session
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 5.3e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.53 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
Chain 1: Iteration: 200 / 2000 [ 10%] (Warmup)
Chain 1: Iteration: 400 / 2000 [ 20%] (Warmup)
Chain 1: Iteration: 600 / 2000 [ 30%] (Warmup)
Chain 1: Iteration: 800 / 2000 [ 40%] (Warmup)
Chain 1: Iteration: 1000 / 2000 [ 50%] (Warmup)
Chain 1: Iteration: 1001 / 2000 [ 50%] (Sampling)
Chain 1: Iteration: 1200 / 2000 [ 60%] (Sampling)
Chain 1: Iteration: 1400 / 2000 [ 70%] (Sampling)
Chain 1: Iteration: 1600 / 2000 [ 80%] (Sampling)
Chain 1: Iteration: 1800 / 2000 [ 90%] (Sampling)
Chain 1: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.293892 seconds (Warm-up)
Chain 1: 0.259832 seconds (Sampling)
Chain 1: 0.553724 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 8.5e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.85 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 1.2e-05 seconds (Warm-up)
Chain 1: 0.003679 seconds (Sampling)
Chain 1: 0.003691 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
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[ 1000 / 1000 ]m.warmup.1 <- map2stan(m, chains = 4, cores = 4, warmup = 1, iter = 1000)starting worker pid=61985 on localhost:11424 at 14:51:10.184
starting worker pid=61999 on localhost:11424 at 14:51:10.479
starting worker pid=62073 on localhost:11424 at 14:51:10.703
starting worker pid=62087 on localhost:11424 at 14:51:10.928
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 5.4e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.54 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1000 [ 0%] (Warmup)
Chain 1: Iteration: 2 / 1000 [ 0%] (Sampling)
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Chain 1: Iteration: 901 / 1000 [ 90%] (Sampling)
Chain 1: Iteration: 1000 / 1000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.000107 seconds (Warm-up)
Chain 1: 0.071876 seconds (Sampling)
Chain 1: 0.071983 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 2).
Chain 2:
Chain 2: Gradient evaluation took 3.8e-05 seconds
Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.38 seconds.
Chain 2: Adjust your expectations accordingly!
Chain 2:
Chain 2:
Chain 2: WARNING: No variance estimation is
Chain 2: performed for num_warmup < 20
Chain 2:
Chain 2: Iteration: 1 / 1000 [ 0%] (Warmup)
Chain 2: Iteration: 2 / 1000 [ 0%] (Sampling)
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Chain 2: Iteration: 801 / 1000 [ 80%] (Sampling)
Chain 2: Iteration: 901 / 1000 [ 90%] (Sampling)
Chain 2: Iteration: 1000 / 1000 [100%] (Sampling)
Chain 2:
Chain 2: Elapsed Time: 0.000101 seconds (Warm-up)
Chain 2: 0.07159 seconds (Sampling)
Chain 2: 0.071691 seconds (Total)
Chain 2:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 3).
Chain 3:
Chain 3: Gradient evaluation took 5.6e-05 seconds
Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.56 seconds.
Chain 3: Adjust your expectations accordingly!
Chain 3:
Chain 3:
Chain 3: WARNING: No variance estimation is
Chain 3: performed for num_warmup < 20
Chain 3:
Chain 3: Iteration: 1 / 1000 [ 0%] (Warmup)
Chain 3: Iteration: 2 / 1000 [ 0%] (Sampling)
Chain 3: Iteration: 101 / 1000 [ 10%] (Sampling)
Chain 3: Iteration: 201 / 1000 [ 20%] (Sampling)
Chain 3: Iteration: 301 / 1000 [ 30%] (Sampling)
Chain 3: Iteration: 401 / 1000 [ 40%] (Sampling)
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Chain 3: Iteration: 801 / 1000 [ 80%] (Sampling)
Chain 3: Iteration: 901 / 1000 [ 90%] (Sampling)
Chain 3: Iteration: 1000 / 1000 [100%] (Sampling)
Chain 3:
Chain 3: Elapsed Time: 0.000113 seconds (Warm-up)
Chain 3: 0.07432 seconds (Sampling)
Chain 3: 0.074433 seconds (Total)
Chain 3:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 4).
Chain 4:
Chain 4: Gradient evaluation took 5.7e-05 seconds
Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.57 seconds.
Chain 4: Adjust your expectations accordingly!
Chain 4:
Chain 4:
Chain 4: WARNING: No variance estimation is
Chain 4: performed for num_warmup < 20
Chain 4:
Chain 4: Iteration: 1 / 1000 [ 0%] (Warmup)
Chain 4: Iteration: 2 / 1000 [ 0%] (Sampling)
Chain 4: Iteration: 101 / 1000 [ 10%] (Sampling)
Chain 4: Iteration: 201 / 1000 [ 20%] (Sampling)
Chain 4: Iteration: 301 / 1000 [ 30%] (Sampling)
Chain 4: Iteration: 401 / 1000 [ 40%] (Sampling)
Chain 4: Iteration: 501 / 1000 [ 50%] (Sampling)
Chain 4: Iteration: 601 / 1000 [ 60%] (Sampling)
Chain 4: Iteration: 701 / 1000 [ 70%] (Sampling)
Chain 4: Iteration: 801 / 1000 [ 80%] (Sampling)
Chain 4: Iteration: 901 / 1000 [ 90%] (Sampling)
Chain 4: Iteration: 1000 / 1000 [100%] (Sampling)
Chain 4:
Chain 4: Elapsed Time: 0.000156 seconds (Warm-up)
Chain 4: 0.062628 seconds (Sampling)
Chain 4: 0.062784 seconds (Total)
Chain 4: There were 3996 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
The largest R-hat is Inf, indicating chains have not mixed.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#r-hatBulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-ess
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.3 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 2e-06 seconds (Warm-up)
Chain 1: 0.001491 seconds (Sampling)
Chain 1: 0.001493 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
Constructing posterior predictions[ 399 / 3996 ]
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[ 3996 / 3996 ]There were 3996 divergent iterations during sampling.
Check the chains (trace plots, n_eff, Rhat) carefully to ensure they are valid.m.warmup.5 <- map2stan(m, chains = 4, cores = 4, warmup = 5, iter = 1000)starting worker pid=62106 on localhost:11424 at 14:51:18.763
starting worker pid=62120 on localhost:11424 at 14:51:18.992
starting worker pid=62134 on localhost:11424 at 14:51:19.212
starting worker pid=62148 on localhost:11424 at 14:51:19.439
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3.5e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.35 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1000 [ 0%] (Warmup)
Chain 1: Iteration: 6 / 1000 [ 0%] (Sampling)
Chain 1: Iteration: 105 / 1000 [ 10%] (Sampling)
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Chain 1: Iteration: 905 / 1000 [ 90%] (Sampling)
Chain 1: Iteration: 1000 / 1000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.000434 seconds (Warm-up)
Chain 1: 0.070503 seconds (Sampling)
Chain 1: 0.070937 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 2).
Chain 2:
Chain 2: Gradient evaluation took 5.7e-05 seconds
Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.57 seconds.
Chain 2: Adjust your expectations accordingly!
Chain 2:
Chain 2:
Chain 2: WARNING: No variance estimation is
Chain 2: performed for num_warmup < 20
Chain 2:
Chain 2: Iteration: 1 / 1000 [ 0%] (Warmup)
Chain 2: Iteration: 6 / 1000 [ 0%] (Sampling)
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Chain 2: Iteration: 905 / 1000 [ 90%] (Sampling)
Chain 2: Iteration: 1000 / 1000 [100%] (Sampling)
Chain 2:
Chain 2: Elapsed Time: 0.000901 seconds (Warm-up)
Chain 2: 0.106665 seconds (Sampling)
Chain 2: 0.107566 seconds (Total)
Chain 2:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 3).
Chain 3:
Chain 3: Gradient evaluation took 4.4e-05 seconds
Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.44 seconds.
Chain 3: Adjust your expectations accordingly!
Chain 3:
Chain 3:
Chain 3: WARNING: No variance estimation is
Chain 3: performed for num_warmup < 20
Chain 3:
Chain 3: Iteration: 1 / 1000 [ 0%] (Warmup)
Chain 3: Iteration: 6 / 1000 [ 0%] (Sampling)
Chain 3: Iteration: 105 / 1000 [ 10%] (Sampling)
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Chain 3: Iteration: 805 / 1000 [ 80%] (Sampling)
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Chain 3: Iteration: 1000 / 1000 [100%] (Sampling)
Chain 3:
Chain 3: Elapsed Time: 0.000515 seconds (Warm-up)
Chain 3: 0.066379 seconds (Sampling)
Chain 3: 0.066894 seconds (Total)
Chain 3:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 4).
Chain 4:
Chain 4: Gradient evaluation took 3.2e-05 seconds
Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.32 seconds.
Chain 4: Adjust your expectations accordingly!
Chain 4:
Chain 4:
Chain 4: WARNING: No variance estimation is
Chain 4: performed for num_warmup < 20
Chain 4:
Chain 4: Iteration: 1 / 1000 [ 0%] (Warmup)
Chain 4: Iteration: 6 / 1000 [ 0%] (Sampling)
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Chain 4: Iteration: 805 / 1000 [ 80%] (Sampling)
Chain 4: Iteration: 905 / 1000 [ 90%] (Sampling)
Chain 4: Iteration: 1000 / 1000 [100%] (Sampling)
Chain 4:
Chain 4: Elapsed Time: 0.000368 seconds (Warm-up)
Chain 4: 0.050052 seconds (Sampling)
Chain 4: 0.05042 seconds (Total)
Chain 4: There were 3452 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupThere were 1 chains where the estimated Bayesian Fraction of Missing Information was low. See
http://mc-stan.org/misc/warnings.html#bfmi-lowExamine the pairs() plot to diagnose sampling problems
The largest R-hat is 18.45, indicating chains have not mixed.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#r-hatBulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-ess
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.3 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 3e-06 seconds (Warm-up)
Chain 1: 0.001561 seconds (Sampling)
Chain 1: 0.001564 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
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[ 3980 / 3980 ]There were 3452 divergent iterations during sampling.
Check the chains (trace plots, n_eff, Rhat) carefully to ensure they are valid.m.warmup.10 <- map2stan(m, chains = 4, cores = 4, warmup = 10, iter = 1000)starting worker pid=62230 on localhost:11424 at 14:51:27.661
starting worker pid=62244 on localhost:11424 at 14:51:27.898
starting worker pid=62258 on localhost:11424 at 14:51:28.119
starting worker pid=62272 on localhost:11424 at 14:51:28.342
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 4.4e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.44 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1000 [ 0%] (Warmup)
Chain 1: Iteration: 11 / 1000 [ 1%] (Sampling)
Chain 1: Iteration: 110 / 1000 [ 11%] (Sampling)
Chain 1: Iteration: 210 / 1000 [ 21%] (Sampling)
Chain 1: Iteration: 310 / 1000 [ 31%] (Sampling)
Chain 1: Iteration: 410 / 1000 [ 41%] (Sampling)
Chain 1: Iteration: 510 / 1000 [ 51%] (Sampling)
Chain 1: Iteration: 610 / 1000 [ 61%] (Sampling)
Chain 1: Iteration: 710 / 1000 [ 71%] (Sampling)
Chain 1: Iteration: 810 / 1000 [ 81%] (Sampling)
Chain 1: Iteration: 910 / 1000 [ 91%] (Sampling)
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 2).
Chain 2:
Chain 2: Gradient evaluation took 5.3e-05 seconds
Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.53 seconds.
Chain 2: Adjust your expectations accordingly!
Chain 2:
Chain 2:
Chain 2: WARNING: No variance estimation is
Chain 2: performed for num_warmup < 20
Chain 2:
Chain 2: Iteration: 1 / 1000 [ 0%] (Warmup)
Chain 2: Iteration: 11 / 1000 [ 1%] (Sampling)
Chain 1: Iteration: 1000 / 1000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.001481 seconds (Warm-up)
Chain 1: 0.942277 seconds (Sampling)
Chain 1: 0.943758 seconds (Total)
Chain 1:
Chain 2: Iteration: 110 / 1000 [ 11%] (Sampling)
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Chain 2: Iteration: 910 / 1000 [ 91%] (Sampling)
Chain 2: Iteration: 1000 / 1000 [100%] (Sampling)
Chain 2:
Chain 2: Elapsed Time: 0.001009 seconds (Warm-up)
Chain 2: 0.912832 seconds (Sampling)
Chain 2: 0.913841 seconds (Total)
Chain 2:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 3).
Chain 3:
Chain 3: Gradient evaluation took 3.5e-05 seconds
Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.35 seconds.
Chain 3: Adjust your expectations accordingly!
Chain 3:
Chain 3:
Chain 3: WARNING: No variance estimation is
Chain 3: performed for num_warmup < 20
Chain 3:
Chain 3: Iteration: 1 / 1000 [ 0%] (Warmup)
Chain 3: Iteration: 11 / 1000 [ 1%] (Sampling)
Chain 3: Iteration: 110 / 1000 [ 11%] (Sampling)
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Chain 3: Iteration: 710 / 1000 [ 71%] (Sampling)
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Chain 3: Iteration: 910 / 1000 [ 91%] (Sampling)
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 4).
Chain 4:
Chain 4: Gradient evaluation took 0.000102 seconds
Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 1.02 seconds.
Chain 4: Adjust your expectations accordingly!
Chain 4:
Chain 4:
Chain 4: WARNING: No variance estimation is
Chain 4: performed for num_warmup < 20
Chain 4:
Chain 4: Iteration: 1 / 1000 [ 0%] (Warmup)
[1] "Error in sampler$call_sampler(args_list[[i]]) : "
[2] " c++ exception (unknown reason)" error occurred during calling the sampler; sampling not doneChain 3: Iteration: 1000 / 1000 [100%] (Sampling)
Chain 3:
Chain 3: Elapsed Time: 0.001239 seconds (Warm-up)
Chain 3: 0.796337 seconds (Sampling)
Chain 3: 0.797576 seconds (Total)
Chain 3: some chains had errors; consider specifying chains = 1 to debughere are whatever error messages were returned[[1]]
Stan model 'log_gdp ~ dnorm(mu, sigma)' does not contain samples.There were 3 chains where the estimated Bayesian Fraction of Missing Information was low. See
http://mc-stan.org/misc/warnings.html#bfmi-lowExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-ess
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3.1e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.31 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 3e-06 seconds (Warm-up)
Chain 1: 0.000961 seconds (Sampling)
Chain 1: 0.000964 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
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[ 2970 / 2970 ]m.warmup.50 <- map2stan(m, chains = 4, cores = 4, warmup = 50, iter = 1000)starting worker pid=62350 on localhost:11424 at 14:51:37.043
starting worker pid=62364 on localhost:11424 at 14:51:37.269
starting worker pid=62378 on localhost:11424 at 14:51:37.512
starting worker pid=62392 on localhost:11424 at 14:51:37.734
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 5.1e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.51 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: There aren't enough warmup iterations to fit the
Chain 1: three stages of adaptation as currently configured.
Chain 1: Reducing each adaptation stage to 15%/75%/10% of
Chain 1: the given number of warmup iterations:
Chain 1: init_buffer = 7
Chain 1: adapt_window = 38
Chain 1: term_buffer = 5
Chain 1:
Chain 1: Iteration: 1 / 1000 [ 0%] (Warmup)
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Chain 1: Iteration: 1000 / 1000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.025593 seconds (Warm-up)
Chain 1: 0.318755 seconds (Sampling)
Chain 1: 0.344348 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 2).
Chain 2:
Chain 2: Gradient evaluation took 5.3e-05 seconds
Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.53 seconds.
Chain 2: Adjust your expectations accordingly!
Chain 2:
Chain 2:
Chain 2: WARNING: There aren't enough warmup iterations to fit the
Chain 2: three stages of adaptation as currently configured.
Chain 2: Reducing each adaptation stage to 15%/75%/10% of
Chain 2: the given number of warmup iterations:
Chain 2: init_buffer = 7
Chain 2: adapt_window = 38
Chain 2: term_buffer = 5
Chain 2:
Chain 2: Iteration: 1 / 1000 [ 0%] (Warmup)
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Chain 2: Iteration: 950 / 1000 [ 95%] (Sampling)
Chain 2: Iteration: 1000 / 1000 [100%] (Sampling)
Chain 2:
Chain 2: Elapsed Time: 0.078622 seconds (Warm-up)
Chain 2: 0.334157 seconds (Sampling)
Chain 2: 0.412779 seconds (Total)
Chain 2:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 3).
Chain 3:
Chain 3: Gradient evaluation took 5.4e-05 seconds
Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.54 seconds.
Chain 3: Adjust your expectations accordingly!
Chain 3:
Chain 3:
Chain 3: WARNING: There aren't enough warmup iterations to fit the
Chain 3: three stages of adaptation as currently configured.
Chain 3: Reducing each adaptation stage to 15%/75%/10% of
Chain 3: the given number of warmup iterations:
Chain 3: init_buffer = 7
Chain 3: adapt_window = 38
Chain 3: term_buffer = 5
Chain 3:
Chain 3: Iteration: 1 / 1000 [ 0%] (Warmup)
Chain 3: Iteration: 51 / 1000 [ 5%] (Sampling)
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Chain 3: Iteration: 850 / 1000 [ 85%] (Sampling)
Chain 3: Iteration: 950 / 1000 [ 95%] (Sampling)
Chain 3: Iteration: 1000 / 1000 [100%] (Sampling)
Chain 3:
Chain 3: Elapsed Time: 0.028215 seconds (Warm-up)
Chain 3: 0.305541 seconds (Sampling)
Chain 3: 0.333756 seconds (Total)
Chain 3:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 4).
Chain 4:
Chain 4: Gradient evaluation took 2.9e-05 seconds
Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.29 seconds.
Chain 4: Adjust your expectations accordingly!
Chain 4:
Chain 4:
Chain 4: WARNING: There aren't enough warmup iterations to fit the
Chain 4: three stages of adaptation as currently configured.
Chain 4: Reducing each adaptation stage to 15%/75%/10% of
Chain 4: the given number of warmup iterations:
Chain 4: init_buffer = 7
Chain 4: adapt_window = 38
Chain 4: term_buffer = 5
Chain 4:
Chain 4: Iteration: 1 / 1000 [ 0%] (Warmup)
Chain 4: Iteration: 51 / 1000 [ 5%] (Sampling)
Chain 4: Iteration: 150 / 1000 [ 15%] (Sampling)
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Chain 4: Iteration: 850 / 1000 [ 85%] (Sampling)
Chain 4: Iteration: 950 / 1000 [ 95%] (Sampling)
Chain 4: Iteration: 1000 / 1000 [100%] (Sampling)
Chain 4:
Chain 4: Elapsed Time: 0.023067 seconds (Warm-up)
Chain 4: 0.273237 seconds (Sampling)
Chain 4: 0.296304 seconds (Total)
Chain 4: There were 1 chains where the estimated Bayesian Fraction of Missing Information was low. See
http://mc-stan.org/misc/warnings.html#bfmi-lowExamine the pairs() plot to diagnose sampling problems
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3.1e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.31 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 3e-06 seconds (Warm-up)
Chain 1: 0.000881 seconds (Sampling)
Chain 1: 0.000884 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
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[ 3800 / 3800 ]m.warmup.100 <- map2stan(m, chains = 4, cores = 4, warmup = 100, iter = 1000)starting worker pid=62470 on localhost:11424 at 14:51:46.342
starting worker pid=62484 on localhost:11424 at 14:51:46.564
starting worker pid=62498 on localhost:11424 at 14:51:46.791
starting worker pid=62512 on localhost:11424 at 14:51:47.012
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 5.6e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.56 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: There aren't enough warmup iterations to fit the
Chain 1: three stages of adaptation as currently configured.
Chain 1: Reducing each adaptation stage to 15%/75%/10% of
Chain 1: the given number of warmup iterations:
Chain 1: init_buffer = 15
Chain 1: adapt_window = 75
Chain 1: term_buffer = 10
Chain 1:
Chain 1: Iteration: 1 / 1000 [ 0%] (Warmup)
Chain 1: Iteration: 100 / 1000 [ 10%] (Warmup)
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Chain 1: Iteration: 900 / 1000 [ 90%] (Sampling)
Chain 1: Iteration: 1000 / 1000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.086368 seconds (Warm-up)
Chain 1: 0.348498 seconds (Sampling)
Chain 1: 0.434866 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 2).
Chain 2:
Chain 2: Gradient evaluation took 6.7e-05 seconds
Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.67 seconds.
Chain 2: Adjust your expectations accordingly!
Chain 2:
Chain 2:
Chain 2: WARNING: There aren't enough warmup iterations to fit the
Chain 2: three stages of adaptation as currently configured.
Chain 2: Reducing each adaptation stage to 15%/75%/10% of
Chain 2: the given number of warmup iterations:
Chain 2: init_buffer = 15
Chain 2: adapt_window = 75
Chain 2: term_buffer = 10
Chain 2:
Chain 2: Iteration: 1 / 1000 [ 0%] (Warmup)
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Chain 2: Iteration: 900 / 1000 [ 90%] (Sampling)
Chain 2: Iteration: 1000 / 1000 [100%] (Sampling)
Chain 2:
Chain 2: Elapsed Time: 0.036724 seconds (Warm-up)
Chain 2: 0.360262 seconds (Sampling)
Chain 2: 0.396986 seconds (Total)
Chain 2:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 3).
Chain 3:
Chain 3: Gradient evaluation took 5.5e-05 seconds
Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.55 seconds.
Chain 3: Adjust your expectations accordingly!
Chain 3:
Chain 3:
Chain 3: WARNING: There aren't enough warmup iterations to fit the
Chain 3: three stages of adaptation as currently configured.
Chain 3: Reducing each adaptation stage to 15%/75%/10% of
Chain 3: the given number of warmup iterations:
Chain 3: init_buffer = 15
Chain 3: adapt_window = 75
Chain 3: term_buffer = 10
Chain 3:
Chain 3: Iteration: 1 / 1000 [ 0%] (Warmup)
Chain 3: Iteration: 100 / 1000 [ 10%] (Warmup)
Chain 3: Iteration: 101 / 1000 [ 10%] (Sampling)
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Chain 3: Iteration: 900 / 1000 [ 90%] (Sampling)
Chain 3: Iteration: 1000 / 1000 [100%] (Sampling)
Chain 3:
Chain 3: Elapsed Time: 0.09803 seconds (Warm-up)
Chain 3: 0.343273 seconds (Sampling)
Chain 3: 0.441303 seconds (Total)
Chain 3:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 4).
Chain 4:
Chain 4: Gradient evaluation took 2.9e-05 seconds
Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.29 seconds.
Chain 4: Adjust your expectations accordingly!
Chain 4:
Chain 4:
Chain 4: WARNING: There aren't enough warmup iterations to fit the
Chain 4: three stages of adaptation as currently configured.
Chain 4: Reducing each adaptation stage to 15%/75%/10% of
Chain 4: the given number of warmup iterations:
Chain 4: init_buffer = 15
Chain 4: adapt_window = 75
Chain 4: term_buffer = 10
Chain 4:
Chain 4: Iteration: 1 / 1000 [ 0%] (Warmup)
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Chain 4: Iteration: 900 / 1000 [ 90%] (Sampling)
Chain 4: Iteration: 1000 / 1000 [100%] (Sampling)
Chain 4:
Chain 4: Elapsed Time: 0.033099 seconds (Warm-up)
Chain 4: 0.251441 seconds (Sampling)
Chain 4: 0.28454 seconds (Total)
Chain 4:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.3 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 3e-06 seconds (Warm-up)
Chain 1: 0.00096 seconds (Sampling)
Chain 1: 0.000963 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
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[ 3600 / 3600 ]m.warmup.500 <- map2stan(m, chains = 4, cores = 4, warmup = 500, iter = 1000)starting worker pid=62590 on localhost:11424 at 14:51:55.888
starting worker pid=62604 on localhost:11424 at 14:51:56.114
starting worker pid=62619 on localhost:11424 at 14:51:56.346
starting worker pid=62635 on localhost:11424 at 14:51:56.592
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 6.5e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.65 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 1000 [ 0%] (Warmup)
[1] "Error in sampler$call_sampler(args_list[[i]]) : "
[2] " c++ exception (unknown reason)" error occurred during calling the sampler; sampling not done
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 2).
Chain 2:
Chain 2: Gradient evaluation took 5.9e-05 seconds
Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.59 seconds.
Chain 2: Adjust your expectations accordingly!
Chain 2:
Chain 2:
Chain 2: Iteration: 1 / 1000 [ 0%] (Warmup)
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Chain 2:
Chain 2: Elapsed Time: 0.194132 seconds (Warm-up)
Chain 2: 0.137121 seconds (Sampling)
Chain 2: 0.331253 seconds (Total)
Chain 2:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 3).
Chain 3:
Chain 3: Gradient evaluation took 3.6e-05 seconds
Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.36 seconds.
Chain 3: Adjust your expectations accordingly!
Chain 3:
Chain 3:
Chain 3: Iteration: 1 / 1000 [ 0%] (Warmup)
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Chain 3:
Chain 3: Elapsed Time: 0.186644 seconds (Warm-up)
Chain 3: 0.136334 seconds (Sampling)
Chain 3: 0.322978 seconds (Total)
Chain 3:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 4).
Chain 4:
Chain 4: Gradient evaluation took 3.7e-05 seconds
Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.37 seconds.
Chain 4: Adjust your expectations accordingly!
Chain 4:
Chain 4:
Chain 4: Iteration: 1 / 1000 [ 0%] (Warmup)
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Chain 4:
Chain 4: Elapsed Time: 0.125488 seconds (Warm-up)
Chain 4: 0.103067 seconds (Sampling)
Chain 4: 0.228555 seconds (Total)
Chain 4: some chains had errors; consider specifying chains = 1 to debughere are whatever error messages were returned[[1]]
Stan model 'log_gdp ~ dnorm(mu, sigma)' does not contain samples.
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 2.9e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.29 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 4e-06 seconds (Warm-up)
Chain 1: 0.000978 seconds (Sampling)
Chain 1: 0.000982 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
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[ 1500 / 1500 ]m.warmup.1000 <- map2stan(m, chains = 4, cores = 4, warmup = 1000, iter = 1000)'iter' less than or equal to 'warmup'. Setting 'iter' to sum of 'iter' and 'warmup' instead (2000).starting worker pid=62714 on localhost:11424 at 14:52:05.225
starting worker pid=62728 on localhost:11424 at 14:52:05.448
starting worker pid=62742 on localhost:11424 at 14:52:05.674
starting worker pid=62756 on localhost:11424 at 14:52:05.901
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 5.3e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.53 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
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Chain 1:
Chain 1: Elapsed Time: 0.362053 seconds (Warm-up)
Chain 1: 0.397191 seconds (Sampling)
Chain 1: 0.759244 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 2).
Chain 2:
Chain 2: Gradient evaluation took 0.000218 seconds
Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 2.18 seconds.
Chain 2: Adjust your expectations accordingly!
Chain 2:
Chain 2:
Chain 2: Iteration: 1 / 2000 [ 0%] (Warmup)
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Chain 2:
Chain 2: Elapsed Time: 0.388013 seconds (Warm-up)
Chain 2: 0.321839 seconds (Sampling)
Chain 2: 0.709852 seconds (Total)
Chain 2:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 3).
Chain 3:
Chain 3: Gradient evaluation took 4.2e-05 seconds
Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.42 seconds.
Chain 3: Adjust your expectations accordingly!
Chain 3:
Chain 3:
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Chain 3:
Chain 3: Elapsed Time: 0.336519 seconds (Warm-up)
Chain 3: 0.318544 seconds (Sampling)
Chain 3: 0.655063 seconds (Total)
Chain 3:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 4).
Chain 4:
Chain 4: Gradient evaluation took 3.1e-05 seconds
Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.31 seconds.
Chain 4: Adjust your expectations accordingly!
Chain 4:
Chain 4:
Chain 4: Iteration: 1 / 2000 [ 0%] (Warmup)
[1] "Error in sampler$call_sampler(args_list[[i]]) : "
[2] " c++ exception (unknown reason)" error occurred during calling the sampler; sampling not done
some chains had errors; consider specifying chains = 1 to debughere are whatever error messages were returned[[1]]
Stan model 'log_gdp ~ dnorm(mu, sigma)' does not contain samples.
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.3 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 3e-06 seconds (Warm-up)
Chain 1: 0.000974 seconds (Sampling)
Chain 1: 0.000977 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
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[ 3000 / 3000 ]precis(m.warmup.1)There were 3996 divergent iterations during sampling.
Check the chains (trace plots, n_eff, Rhat) carefully to ensure they are valid.precis(m.warmup.5)There were 3452 divergent iterations during sampling.
Check the chains (trace plots, n_eff, Rhat) carefully to ensure they are valid.precis(m.warmup.10)precis(m.warmup.50)precis(m.warmup.100)precis(m.warmup.500)precis(m.warmup.1000)Hard
8H1
mp <- map2stan(
alist(
a ~ dnorm(0, 1),
b ~ dcauchy(0, 1)
),
data = list(y = 1),
start = list(a = 0, b = 0),
iter = 1e4, warmup = 100, WAIC = FALSE
)
SAMPLING FOR MODEL 'a ~ dnorm(0, 1)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 7e-06 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.07 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: There aren't enough warmup iterations to fit the
Chain 1: three stages of adaptation as currently configured.
Chain 1: Reducing each adaptation stage to 15%/75%/10% of
Chain 1: the given number of warmup iterations:
Chain 1: init_buffer = 15
Chain 1: adapt_window = 75
Chain 1: term_buffer = 10
Chain 1:
Chain 1: Iteration: 1 / 10000 [ 0%] (Warmup)
Chain 1: Iteration: 101 / 10000 [ 1%] (Sampling)
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Chain 1: Iteration: 10000 / 10000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.003766 seconds (Warm-up)
Chain 1: 0.318112 seconds (Sampling)
Chain 1: 0.321878 seconds (Total)
Chain 1: There were 5 transitions after warmup that exceeded the maximum treedepth. Increase max_treedepth above 10. See
http://mc-stan.org/misc/warnings.html#maximum-treedepth-exceededExamine the pairs() plot to diagnose sampling problems
SAMPLING FOR MODEL 'a ~ dnorm(0, 1)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 5e-06 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 3e-06 seconds (Warm-up)
Chain 1: 0.001385 seconds (Sampling)
Chain 1: 0.001388 seconds (Total)
Chain 1: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essformula8m3 <- alist(
log_gdp ~ dnorm( mu , sigma ) ,
mu <- a + bR*rugged + bA*cont_africa + bAR*rugged*cont_africa ,
a ~ dnorm(0,100),
bR ~ dnorm(0,10),
bA ~ dnorm(0,10),
bAR ~ dnorm(0,10),
sigma ~ dcauchy(0,2)
)
m8m3.w1 <- map2stan(formula8m3, data=dd.trim, iter=1001, warmup=1)recompiling to avoid crashing R session
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3.1e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.31 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1001 [ 0%] (Warmup)
Chain 1: Iteration: 2 / 1001 [ 0%] (Sampling)
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Chain 1: Iteration: 1001 / 1001 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.000836 seconds (Warm-up)
Chain 1: 0.073573 seconds (Sampling)
Chain 1: 0.074409 seconds (Total)
Chain 1: There were 1000 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3.2e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.32 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 4e-06 seconds (Warm-up)
Chain 1: 0.000927 seconds (Sampling)
Chain 1: 0.000931 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
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[ 1000 / 1000 ]There were 1000 divergent iterations during sampling.
Check the chains (trace plots, n_eff, Rhat) carefully to ensure they are valid.precis(m8m3.w1)#awfull results, n_eff=1There were 1000 divergent iterations during sampling.
Check the chains (trace plots, n_eff, Rhat) carefully to ensure they are valid.m8m3.w10 <- map2stan(formula8m3, data=dd.trim, iter=1010, warmup=10)recompiling to avoid crashing R session
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3.4e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.34 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1010 [ 0%] (Warmup)
Chain 1: Iteration: 11 / 1010 [ 1%] (Sampling)
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Chain 1: Iteration: 919 / 1010 [ 90%] (Sampling)
Chain 1: Iteration: 1010 / 1010 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.001568 seconds (Warm-up)
Chain 1: 0.699901 seconds (Sampling)
Chain 1: 0.701469 seconds (Total)
Chain 1: There were 1 chains where the estimated Bayesian Fraction of Missing Information was low. See
http://mc-stan.org/misc/warnings.html#bfmi-lowExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-ess
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 2.1e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.21 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 4e-06 seconds (Warm-up)
Chain 1: 0.000727 seconds (Sampling)
Chain 1: 0.000731 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
Constructing posterior predictions[ 100 / 1000 ]
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[ 1000 / 1000 ]precis(m8m3.w10)#not so awfull results, n_eff=~100..200, troubles with estimates bA & sigmaplot(m8m3.w10)
m8m3.w100 <- map2stan(formula8m3, data=dd.trim, iter=1100, warmup=100)recompiling to avoid crashing R session
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 2.9e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.29 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: There aren't enough warmup iterations to fit the
Chain 1: three stages of adaptation as currently configured.
Chain 1: Reducing each adaptation stage to 15%/75%/10% of
Chain 1: the given number of warmup iterations:
Chain 1: init_buffer = 15
Chain 1: adapt_window = 75
Chain 1: term_buffer = 10
Chain 1:
Chain 1: Iteration: 1 / 1100 [ 0%] (Warmup)
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Chain 1: Iteration: 1090 / 1100 [ 99%] (Sampling)
Chain 1: Iteration: 1100 / 1100 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.065957 seconds (Warm-up)
Chain 1: 0.280669 seconds (Sampling)
Chain 1: 0.346626 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 2.1e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.21 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 2.1e-05 seconds (Warm-up)
Chain 1: 0.003718 seconds (Sampling)
Chain 1: 0.003739 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
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[ 1000 / 1000 ]precis(m8m3.w100)#enough of warmupplot(m8m3.w100)
m8m3.w500 <- map2stan(formula8m3, data=dd.trim, iter=1500, warmup=500)recompiling to avoid crashing R session
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.3 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 1500 [ 0%] (Warmup)
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Chain 1: Iteration: 1400 / 1500 [ 93%] (Sampling)
Chain 1: Iteration: 1500 / 1500 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.256748 seconds (Warm-up)
Chain 1: 0.323527 seconds (Sampling)
Chain 1: 0.580275 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 5.8e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.58 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 1.8e-05 seconds (Warm-up)
Chain 1: 0.003977 seconds (Sampling)
Chain 1: 0.003995 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
Constructing posterior predictions[ 100 / 1000 ]
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[ 1000 / 1000 ]precis(m8m3.w500)#"wasted" warmupplot(m8m3.w500)
m8m3.w1k <- map2stan(formula8m3, data=dd.trim, iter=2000, warmup=1000)recompiling to avoid crashing R session
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 3.2e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.32 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
Chain 1: Iteration: 200 / 2000 [ 10%] (Warmup)
Chain 1: Iteration: 400 / 2000 [ 20%] (Warmup)
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Chain 1: Iteration: 1001 / 2000 [ 50%] (Sampling)
Chain 1: Iteration: 1200 / 2000 [ 60%] (Sampling)
Chain 1: Iteration: 1400 / 2000 [ 70%] (Sampling)
Chain 1: Iteration: 1600 / 2000 [ 80%] (Sampling)
Chain 1: Iteration: 1800 / 2000 [ 90%] (Sampling)
Chain 1: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.272828 seconds (Warm-up)
Chain 1: 0.30769 seconds (Sampling)
Chain 1: 0.580518 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'log_gdp ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 5.8e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.58 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 1.1e-05 seconds (Warm-up)
Chain 1: 0.001692 seconds (Sampling)
Chain 1: 0.001703 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
Constructing posterior predictions[ 100 / 1000 ]
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[ 1000 / 1000 ]precis(m8m3.w1k)#"wasted" warmupplot(m8m3.w1k)
precis(mp)pairs(mp)
2H2
samples <- extract.samples(mp)
hist(samples$a)
hist(samples$b)
library(dplyr)
Attaching package: ‘dplyr’
The following objects are masked from ‘package:stats’:
filter, lag
The following objects are masked from ‘package:base’:
intersect, setdiff, setequal, unionm5.1 <- map2stan(
alist(
Divorce ~ dnorm( mu , sigma ) ,
mu <- a + bA * MedianAgeMarriage_s ,
a ~ dnorm( 10 , 10 ) ,
bA ~ dnorm( 0 , 1 ) ,
sigma ~ dcauchy(0 , 2)
),
data = select(d, Divorce, MedianAgeMarriage_s) )
SAMPLING FOR MODEL 'Divorce ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 1.3e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.13 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
Chain 1: Iteration: 200 / 2000 [ 10%] (Warmup)
Chain 1: Iteration: 400 / 2000 [ 20%] (Warmup)
Chain 1: Iteration: 600 / 2000 [ 30%] (Warmup)
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Chain 1: Iteration: 1000 / 2000 [ 50%] (Warmup)
Chain 1: Iteration: 1001 / 2000 [ 50%] (Sampling)
Chain 1: Iteration: 1200 / 2000 [ 60%] (Sampling)
Chain 1: Iteration: 1400 / 2000 [ 70%] (Sampling)
Chain 1: Iteration: 1600 / 2000 [ 80%] (Sampling)
Chain 1: Iteration: 1800 / 2000 [ 90%] (Sampling)
Chain 1: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.047693 seconds (Warm-up)
Chain 1: 0.037122 seconds (Sampling)
Chain 1: 0.084815 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'Divorce ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 8e-06 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.08 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 2e-06 seconds (Warm-up)
Chain 1: 0.000721 seconds (Sampling)
Chain 1: 0.000723 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
Constructing posterior predictions[ 100 / 1000 ]
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[ 1000 / 1000 ]precis(m5.1)plot(m5.1)
pairs(m5.1)
m5.2 <- map2stan(
alist(
Divorce ~ dnorm( mu , sigma ) ,
mu <- a + bR * Marriage_s ,
a ~ dnorm( 10 , 10 ) ,
bR ~ dnorm( 0 , 1 ) ,
sigma ~ dcauchy( 0 , 2 )
) , data = select(d, Divorce, Marriage_s) )
SAMPLING FOR MODEL 'Divorce ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 1.4e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.14 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
Chain 1: Iteration: 200 / 2000 [ 10%] (Warmup)
Chain 1: Iteration: 400 / 2000 [ 20%] (Warmup)
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Chain 1: Iteration: 1000 / 2000 [ 50%] (Warmup)
Chain 1: Iteration: 1001 / 2000 [ 50%] (Sampling)
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Chain 1: Iteration: 1600 / 2000 [ 80%] (Sampling)
Chain 1: Iteration: 1800 / 2000 [ 90%] (Sampling)
Chain 1: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.044218 seconds (Warm-up)
Chain 1: 0.062542 seconds (Sampling)
Chain 1: 0.10676 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'Divorce ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 1.2e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 3e-06 seconds (Warm-up)
Chain 1: 0.000823 seconds (Sampling)
Chain 1: 0.000826 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
Constructing posterior predictions[ 100 / 1000 ]
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[ 1000 / 1000 ]precis(m5.2)plot(m5.2)
pairs(m5.2)
m5.3 <- map2stan(
alist(
Divorce ~ dnorm( mu , sigma ) ,
mu <- a + bR*Marriage_s + bA*MedianAgeMarriage_s ,
a ~ dnorm( 10 , 10 ) ,
bR ~ dnorm( 0 , 1 ) ,
bA ~ dnorm( 0 , 1 ) ,
sigma ~ dcauchy( 0 , 2)
) ,
data = select(d, Divorce, MedianAgeMarriage_s, Marriage_s)
)
SAMPLING FOR MODEL 'Divorce ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 1.6e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.16 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
Chain 1: Iteration: 200 / 2000 [ 10%] (Warmup)
Chain 1: Iteration: 400 / 2000 [ 20%] (Warmup)
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Chain 1: Iteration: 1000 / 2000 [ 50%] (Warmup)
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Chain 1: Iteration: 1800 / 2000 [ 90%] (Sampling)
Chain 1: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.067842 seconds (Warm-up)
Chain 1: 0.071936 seconds (Sampling)
Chain 1: 0.139778 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'Divorce ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 9e-06 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.09 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 4e-06 seconds (Warm-up)
Chain 1: 0.002628 seconds (Sampling)
Chain 1: 0.002632 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
Constructing posterior predictions[ 100 / 1000 ]
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[ 1000 / 1000 ]precis(m5.3)plot(m5.3)
pairs(m5.3)
8H3
## R code 8.21
N <- 100 # number of individuals
height <- rnorm(N,10,2) # sim total height of each
leg_prop <- runif(N,0.4,0.5) # leg as proportion of height
leg_left <- leg_prop*height + # sim left leg as proportion + error
rnorm( N , 0 , 0.02 )
leg_right <- leg_prop*height + # sim right leg as proportion + error
rnorm( N , 0 , 0.02 )
# combine into data frame
d <- data.frame(height,leg_left,leg_right)
m5.8s <- map2stan(
alist(
height ~ dnorm( mu , sigma ) ,
mu <- a + bl*leg_left + br*leg_right ,
a ~ dnorm( 10 , 100 ) ,
bl ~ dnorm( 2 , 10 ) ,
br ~ dnorm( 2 , 10 ) ,
sigma ~ dcauchy( 0 , 1 )
) ,
data=d, chains=4,
start=list(a=10,bl=0,br=0,sigma=1) )
m5.8s2 <- map2stan(
alist(
height ~ dnorm( mu , sigma ) ,
mu <- a + bl*leg_left + br*leg_right ,
a ~ dnorm( 10 , 100 ) ,
bl ~ dnorm( 2 , 10 ) ,
br ~ dnorm( 2 , 10 ) & T[0,] ,
sigma ~ dcauchy( 0 , 1 )
) ,
data=d, chains=4,
start=list(a=10,bl=0,br=0,sigma=1) )
SAMPLING FOR MODEL 'height ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 2.1e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.21 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
Chain 1: Iteration: 200 / 2000 [ 10%] (Warmup)
Chain 1: Iteration: 400 / 2000 [ 20%] (Warmup)
Chain 1: Iteration: 600 / 2000 [ 30%] (Warmup)
Chain 1: Iteration: 800 / 2000 [ 40%] (Warmup)
Chain 1: Iteration: 1000 / 2000 [ 50%] (Warmup)
Chain 1: Iteration: 1001 / 2000 [ 50%] (Sampling)
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Chain 1: Iteration: 1800 / 2000 [ 90%] (Sampling)
Chain 1: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 3.99865 seconds (Warm-up)
Chain 1: 5.00988 seconds (Sampling)
Chain 1: 9.00854 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'height ~ dnorm(mu, sigma)' NOW (CHAIN 2).
Chain 2:
Chain 2: Gradient evaluation took 1.3e-05 seconds
Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.13 seconds.
Chain 2: Adjust your expectations accordingly!
Chain 2:
Chain 2:
Chain 2: Iteration: 1 / 2000 [ 0%] (Warmup)
Chain 2: Iteration: 200 / 2000 [ 10%] (Warmup)
Chain 2: Iteration: 400 / 2000 [ 20%] (Warmup)
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Chain 2: Iteration: 1001 / 2000 [ 50%] (Sampling)
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Chain 2: Iteration: 1800 / 2000 [ 90%] (Sampling)
Chain 2: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 2:
Chain 2: Elapsed Time: 3.58251 seconds (Warm-up)
Chain 2: 4.99257 seconds (Sampling)
Chain 2: 8.57507 seconds (Total)
Chain 2:
SAMPLING FOR MODEL 'height ~ dnorm(mu, sigma)' NOW (CHAIN 3).
Chain 3:
Chain 3: Gradient evaluation took 1.3e-05 seconds
Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.13 seconds.
Chain 3: Adjust your expectations accordingly!
Chain 3:
Chain 3:
Chain 3: Iteration: 1 / 2000 [ 0%] (Warmup)
Chain 3: Iteration: 200 / 2000 [ 10%] (Warmup)
Chain 3: Iteration: 400 / 2000 [ 20%] (Warmup)
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Chain 3: Iteration: 800 / 2000 [ 40%] (Warmup)
Chain 3: Iteration: 1000 / 2000 [ 50%] (Warmup)
Chain 3: Iteration: 1001 / 2000 [ 50%] (Sampling)
Chain 3: Iteration: 1200 / 2000 [ 60%] (Sampling)
Chain 3: Iteration: 1400 / 2000 [ 70%] (Sampling)
Chain 3: Iteration: 1600 / 2000 [ 80%] (Sampling)
Chain 3: Iteration: 1800 / 2000 [ 90%] (Sampling)
Chain 3: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 3:
Chain 3: Elapsed Time: 3.24065 seconds (Warm-up)
Chain 3: 5.13321 seconds (Sampling)
Chain 3: 8.37386 seconds (Total)
Chain 3:
SAMPLING FOR MODEL 'height ~ dnorm(mu, sigma)' NOW (CHAIN 4).
Chain 4:
Chain 4: Gradient evaluation took 1.8e-05 seconds
Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.18 seconds.
Chain 4: Adjust your expectations accordingly!
Chain 4:
Chain 4:
Chain 4: Iteration: 1 / 2000 [ 0%] (Warmup)
Chain 4: Iteration: 200 / 2000 [ 10%] (Warmup)
Chain 4: Iteration: 400 / 2000 [ 20%] (Warmup)
Chain 4: Iteration: 600 / 2000 [ 30%] (Warmup)
Chain 4: Iteration: 800 / 2000 [ 40%] (Warmup)
Chain 4: Iteration: 1000 / 2000 [ 50%] (Warmup)
Chain 4: Iteration: 1001 / 2000 [ 50%] (Sampling)
Chain 4: Iteration: 1200 / 2000 [ 60%] (Sampling)
Chain 4: Iteration: 1400 / 2000 [ 70%] (Sampling)
Chain 4: Iteration: 1600 / 2000 [ 80%] (Sampling)
Chain 4: Iteration: 1800 / 2000 [ 90%] (Sampling)
Chain 4: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 4:
Chain 4: Elapsed Time: 3.24088 seconds (Warm-up)
Chain 4: 4.24708 seconds (Sampling)
Chain 4: 7.48796 seconds (Total)
Chain 4: There were 93 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupThere were 799 transitions after warmup that exceeded the maximum treedepth. Increase max_treedepth above 10. See
http://mc-stan.org/misc/warnings.html#maximum-treedepth-exceededExamine the pairs() plot to diagnose sampling problems
SAMPLING FOR MODEL 'height ~ dnorm(mu, sigma)' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 1.3e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.13 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 2e-06 seconds (Warm-up)
Chain 1: 0.00082 seconds (Sampling)
Chain 1: 0.000822 seconds (Total)
Chain 1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupExamine the pairs() plot to diagnose sampling problems
Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-essTail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-essComputing WAIC
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[ 4000 / 4000 ]There were 93 divergent iterations during sampling.
Check the chains (trace plots, n_eff, Rhat) carefully to ensure they are valid.coeftab(m5.1, m5.2, m5.3) m5.1 m5.2 m5.3
a 9.69 9.69 9.70
bA -1.04 NA -1.13
sigma 1.50 1.73 1.52
bR NA 0.65 -0.12
nobs 50 50 50compare(m5.1, m5.2, m5.3)NAprecis(m5.8s)pairs(m5.8s)
precis(m5.8s2)There were 93 divergent iterations during sampling.
Check the chains (trace plots, n_eff, Rhat) carefully to ensure they are valid.pairs(m5.8s2)
8H4
compare(m5.8s, m5.8s2)8H5
# using models from 8H3
WAIC(m5.8s) # pWAIC = 3.2937[1] 222.936
attr(,"lppd")
[1] -107.7326
attr(,"pWAIC")
[1] 3.735358
attr(,"se")
[1] 12.97404WAIC(m5.8s2) # pWAIC = 2.819[1] 222.787
attr(,"lppd")
[1] -107.7871
attr(,"pWAIC")
[1] 3.606366
attr(,"se")
[1] 12.80438# Effective number of parameters for the second model is smaller.
# Intuitively it is smaller because we restricted "freedom" of the 'br' coefficient. This parameter couldn't be negative for the second model, while the probability of having big values is still very small as for the first model. Thus overall freedom of the model declined.
# More formally, pWAIC is defined as sum of variance of the points likelihood, thus the second model has smaller variance of data likelihood(==> it's 'more restricted')
#same story with effective numbet of parameters for DIC
DIC(m5.8s) # pD=3.9[1] 222.8214
attr(,"pD")
[1] 3.847699DIC(m5.8s2) # pD=3.4[1] 222.7807
attr(,"pD")
[1] 3.8160938H6
population <- c(10, 60, 20, 100, 30)
n_islands <- length(population)
n_trials <- 1e+5
positions <- rep(0, n_trials)
curr_pos <- 1
for(i in 1:n_trials){
positions[i] <- curr_pos
next_pos <- ifelse(runif(1)<0.5, -1, 1) + curr_pos
if (next_pos <= 0){
next_pos <- n_islands
} else if (next_pos > n_islands){
next_pos <- 1
}
p_ratio <- population[next_pos] / population[curr_pos]
if ( runif(1) < p_ratio ){
curr_pos <- next_pos
}
}
hist(positions)
table(positions)/n_trialspositions
1 2 3 4 5
0.04454 0.27883 0.09030 0.45101 0.13532 population/sum(population)[1] 0.04545455 0.27272727 0.09090909 0.45454545 0.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